Result
Square Root Calculator: Principal Real Root
Find the principal square root of a non-negative number.
Find the non-negative square root of a number and distinguish that result from the two solutions of a squared equation.
Formula: For x ≥ 0, √x is the non-negative number r such that r × r = x.
√144 = 12 because 12 × 12 = 144. The equation r² = 144 has two solutions, +12 and −12; the principal square root is 12.
This calculator returns real roots only. Negative input is rejected, and irrational results are rounded to limited significant digits.
Method and examples
A square root can be smaller than its input
For numbers above one, the positive square root is smaller than the input: √81 = 9. Between zero and one, the root is larger: √0.04 = 0.2. Zero and one are unchanged. These checks help catch a misplaced decimal point.
Understand approximate roots
For a non-perfect square such as 2, the decimal expansion does not terminate. The displayed result is an approximation, not an exact symbolic simplification. Squaring a rounded display may differ slightly from the original number. Keep sufficient precision in intermediate work and round only the final answer.
Use square units consistently
If a square has area 225 cm², its side length is √225 = 15 cm. Taking a square root changes square units into length units. It does not convert centimetres to metres. To apply the same idea to a different area unit, convert the area first or retain its matching length unit.
Know which problems need a different method
A negative real number has no real square root; complex arithmetic uses the imaginary unit instead. This calculator does not solve general quadratic equations or simplify radicals into exact algebraic form. For a right triangle, calculate the sum of squared perpendicular sides before taking the root, or use the Pythagorean tool.
Maintained by: Muhammad Bhutto · 2026-10-17 · Methodology · Report an error